We develop the contact singularity theory for singularly perturbed (or ‘slow–fast’) vector fields of the general form \(z' = H(z,\varepsilon )\), \(z\in {\mathbb {R}}^n\) and \(0 < \varepsilon \ll 1\). Our main result is the derivation of computable, coordinate-independent defining equations for contact singularities under an assumption that the leading-order term of the vector field admits a suitable factorization. This factorization can in turn be computed explicitly in a wide variety of applications. We demonstrate these computable criteria by locating contact folds and, for the first time, contact cusps in general slow–fast models of biochemical oscillators and the Yamada model for self-pulsating lasers.

我们针对一般形式为\（z'= H（z，\ varepsilon）\），\（z \ in {\ mathbb {R}}的奇异摄动（或“慢-快”）向量场开发了接触奇点理论^ n \）和\（0 <\ varepsilon \ ll 1 \）。我们的主要结果是在矢量场的前导项允许适当分解的前提下，推导了用于接触奇点的可计算，独立于坐标的定义方程。反过来，可以在各种应用中显式计算该因式分解。我们通过定位接触折叠以及首次在生物化学振荡器的一般慢速-快速模型和用于自脉冲激光器的Yamada模型中的接触尖峰来证明这些可计算的标准。